2004-2005 Graduate Course Descriptions

CORE COURSES

1. Lebesgue integration, measure theory, convergence theorems, the Riesz representation theorem, Fubini’s theorem, complex measures.
2. L^p-spaces, density of continuous functions, Hilbert space, weak and strong topologies, integral operators.
3. Inequalities.
4. Bounded linear operators and functionals. Hahn-Banach theorem, open-mapping theorem, closed graph theorem, uniform boundedness principle.
5. Schwartz space, introduction to distributions, Fourier transforms on the circle and the line (Schwartz space and L₂ ).
6. Spectral theorem for bounded normal operators.

1. Review of elementary properties of holomorphic functions. Cauchy~Rs integral formula, Taylor and Laurent series, residue calculus.
2. Harmonic functions. Poisson's integral formula and Dirichlet's problem.
3. Conformal mapping, Riemann mapping theorem.
4. Analytic continuation, monodromy theorem, elementary Riemann surfaces.
5. Elliptic functions, the modular function, little Picard theorem.

1. Linear Algebra. Students will be expected to have a good grounding in linear algebra, vector spaces, dual spaces, direct sum, linear transformations and matrices, determinants, eigenvectors, minimal polynomials, Jordan canonical form, Cayley-Hamilton theorem, symmetric, alternating and Hermitian forms, polar decomposition.
2. Group Theory. Isomorphism theorems, group actions, Jordan-Hölder theorem, Sylow theorems, direct and semidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups, nilpotent and solvable groups, generators and relations.
3. Ring Theory. Rings, ideals, rings of fractions and localization, factorization theory, Noetherian rings, Hilbert basis theorem, invariant theory,Hilbert Nullstellensatz, primary decomposition, affine algebraic varieties.
4. Modules. Modules and algebras over a ring, tensor products, modules over a principal ideal domain, applications to linear algebra, structure of semisimple algebras, application to representation theory of finite groups.
5. Fields. Algebraic and transcendental extensions, normal and separable extensions, fundamental theorem of Galois theory, solution of equations by radicals.

1. Point set topology: Topological spaces and continuous functions, connectedness and compactness, countability and separation axioms.
2. Homotopy: Fundamental group, Van Kampen theorem, Brouwer's theorem for the 2-disk. Homotopy of spaces and maps, higher homotopy groups.
3. The language of category theory.
4. Covering theory, universal coverings.
5. Homology: Simplicial and singular homology, homotopy invariance, exact sequences, Mayer-Vietoris, excision, Brouwer's theorem for the n-disk, degrees of maps, CW-complexes, Euler characteristic, a word about the classification of surfaces.
6. Cohomology: Cohomology groups, cup products, cohomology with coefficients.
7. Topological manifolds: Orientation, fundamental class, Poincare duality.

2004-2005 GRADUATE COURSES

The multidimensional Fourier transform. Reconstruction from Fourier transform samples, Nyquist's theorem, Poisson summation.

The Radon transform. Reconstruction from Radon transform samples.

The Bloch Equation in Magnetic Resonance Imaging; connection to the analytic development of the Inverse Scattering Transform (also used in the exact solution of integrable nonlinear partial differential equations).

Current Density Impedance Imaging.

(time permitting) Partial differential equations techniques for Ultrasound Imaging and Electrical Impedance Tomography.

This course is concerned with the basic aspects of C*-algebras. During the first half of the semester, the lectures will be devoted to a systematic investigation of the finite-dimensional case; namely, the theory of complex matrices. Through many simple concrete examples we may describe various phenomena arising from the interplay of normed structure, order structure and algebraic structure. Later in the semester, we will continue to explore more striking facts pertinent to the infinite-dimensional case. Based on the background of students, various topics of related interest will be pursued.

Graduate students taking this course for credit will be required to present a written report on a topic pertinent to their mathematical backgrounds.

MAT 1051HF (MAT 468H1F)
ORDINARY DIFFERENTIAL EQUATIONS
M. Goldstein

1. Theorem on existance and uniqueness of solutions:
   • Vector fields and ODE's,
   • Contracting map principle,
   • Theorem on existence and uniqueness,
   • Differentiability of solutions with respect to initial conditions and parameters,
   • Flow defined by an ODE.
2. Linear systems:
   • Exponentials of linear operators in Euclidean space,
   • Canonical form of a linear operator acting in real Euclidean space,
   • Phase portrets of a linear system in plane.
3. Stability of solutions:
   • Definition of stability and asymptotic stability,
   • Theorems on stability and instability in terms of eigenvalues of the linearized system.
4. Differentiable manifolds:
   • Local charts,
   • Tangent space,
   • Vector fields and ODE's on differentiable manifolds.
5. Differential forms and Stokes' formula:
   • Exterior forms on $\IR^n$,
   • Differential forms on manifolds,
   • Exterior product and differentiation of differential forms,
   • Stokes' formula,
   • Stokes' formula in $\IR^3$,
   • Vortex field of a vector field on $\IR^{2n+1}$,
   • Hydrodynamical lemma
6. Newton equations and Hamiltonian systems:
   • Newton equations for N-body problem,
   • Angular momentum and Kepler's laws for two body problem,
   • Hamiltonian system for N-body problem,
   • Liouville's Theorem on conservation of the volume in the phase space.
7. Poisson's brackets of Hamiltonian functions:
   • Poisson's bracket of vector fields,
   • The Jacobi identity,
   • Poisson's bracket of Hamiltonian functions,
   • Conservation laws for a Hamiltonian system.
8. Liouville theorem on integrable systems:
   • Action of the flows defined by 1st integrals,
   • Discrete subgroups in $\IR^n$,
   • Hamiltonian system on unvariant tori
9. Canonical formulism:
   • The vortex form of a Hamiltonian system,
   • Poincare-Cartan's integral invariant,
   • Canonical change of variables,
   • Generating function,
   • Hamilton-Jacobi equations.
10. Action-angle variables:
    • Action-angle variables in the case of 1 degree of freedom,
    • Action-angle variables of an integrable system.
11. Kolmogorov's theorem:
    • Theorem on averages,
    • Averages of small perturbations,
    • Statement of Kolmogorov's theorem.

This one-semester course is a basic introduction the the theory of partial differential equations. It is intended to help students get up to speed for the topics in PDEs course being taught in the spring: "Asymptotic Methods for PDEs", "Nonlinear Schroedinger Equations", "General Relativity".

Prerequisites: either you should have taken a course on measure and integration or you should be willing to suspend disbelief when presented with things like the Cauchy-Scwartz inequality and the Dominated Convergence Theorm.

Textbook: "Partial Differential Equations" by Lawrence C. Evans

Course syllabus:

Chapter 2: Four Important Linear PDES
  2.1 Transport equation
  2.2 Laplace's equation
  2.3 Heat equation
  2.4 Wave equation

Chapter 3: Nonlinear First-Order PDE
  3.2 Characteristics
  3.4 Conservation Laws

Chapter 4: Other ways to represent solutions
  4.1 Separation of variables
  4.2 similarity solutions
  4.3 Transform methods

Chapter 6: Second-order elliptic equations
  6.1 definitions
  6.2 existence of weak solutions
  6.3.1 regularity 
  6.4 maximum principles
  6.5.1 eigenvalues and eigenfunctions 

Chapter 7: Linear evolution equations
  7.1 second-order parabolic (if time remains)  

When possible, I will include computer demonstrations and will provide matlab code for computationally-inclined students to play with.

MAT 1075HF
INTRODUCTION TO MICROLOCAL ANALYSIS
V. Ivrii

1. Pseudodifferential operators:
   • Symbols, Quantization, Calculus.
   • Oscillatory Front sets, coherent states, microlocalization.
   • L²-estimates, Garding inequalities.
2. Fourier Integral Operators:
   • Oscillatory solutions, relation to classical dynamics.
   • Lagrangian distributions, phase functions and amplitudes, Lagrangian manifolds.
   • Canonical graphs and Fourier Integral Operators.
   • Metaplectic operators.
   • Oscillatory solutions near and beyond caustics.
   • Maslov Canonical Operator.
3. Propagation of singularities.
   • Fourier Integral Operators Approach.
   • Heizenberg approach.
   • Coherent States approach.
   • Energy estimates approach.
4. Applications to Spectral Asymptotics.
   • Tauberian Approach.
   • Poisson Relations, Spectrum and Length Spectrum.
   • Sharp spectral asymptotics.

Definitions, structure, and representation theorems for Abelian Categories. Resolutions by projectives, injectives and cohomology theories. Hochschild Cohomology, Gerstenhaber Algebras, Super Poisson and Super Lie algebras. Triangulated Categories, exact functors Fourier-Mukai transforms and other equivalences. Stable theories: Tate cohomolgy, periodicity. The homological mirror conjectures.

Prerequisite:
Core course in Algebra required, interest in algebraic geometry helpful.

MAT 1120HS
LIE GROUPS AND LIE ALGEBRAS
J. Scherk

Lie groups, closed subgroups, homogeneous spaces. Lie algebras, Lie's theorem, solvable and nilpotent Lie algebras; semi-simple Lie algebras, root systems, classification.

References:
Bourbaki: Lie Groups and Lie Algebras
Carter, Segal, Macdonald: Lectures on Lie Groups and Lie Algebras
Knapp: Lie Groups Beyond an Introduction
Serre: Lie Groups and Lie Algebras

MAT 1122HF
HAMILTONIAN SYSTEMS ON LIE GROUPS AND THEIR HOMOGENEOUS SPACES
V. Jurdjevic

This course would assume some familiarity with Lie groups and differentiable manifolds and would be based on my recent manuscript dealing with variational problems in control and quantum control, Riemannian and sub-Riemannian geometry, and other problems of applied mathematics. In a sense, the course will be a blend of optimal control theory, symplectic and differential geometry and mathematical physics. I would like to include the following topics:

1. Differential systems, families of vector fields and their accessibilty properties.
2. The Maximum Principle of optimality and the associated Hamiltonian formalism.
3. The left-invariant symplectic form of the cotangent bundle of a Lie group, coadjoint orbits, and the Poisson structure of the dual of a Lie algebra.
4. Symmetric Riemannian spaces seen through the formalism of the sub-Riemannian structure of the associated principal bundles.
5. Kirchhoff's Elastic problems, the Euler-Griffiths problem and the equations of the rigid body.
6. Integrable Hamiltonian systems on Lie groups, Kowalewski-Lyapunov methods and single valued solutions.
7. Complex symplectic geometry, real forms and applications to geometry and mechanics.
8. Optimal problems on the unitary group U_n and the problems of quantum control.
9. Infinite dimensional integrable Hamiltonian systems, the non-linear Schroedinger's equation and solitons.

Basic notions of algebraic geometry, with emphasis on geometry

Algebraic topics: fields, rings, ideals and moduli, dimension theory, algebraic aspects of infinitesimal notions, valuations.

Geometric topics: affine and projective varieties, dimension and intersection theory, curves and surfaces, schemes, varieties over the complex and real numbers.

Textbook:
Shafarevich, I.R.: Basic algebraic geometry (Vols. 1 and 2), second edition, Springer-Verlag, Berlin, 1994.

A reference for algebraic topics:
Shafarevich, I.R.: Basic notions of algebra, Springer-Verlag, Berlin, 1997.

MAT 1191HF
TOPICS IN ALGEBRAIC GEOMETRY
C. Consani

This course will focus on two main topics in algebraic geometry:
- pure motives
- Grothendieck groups attached to an algebraic scheme.

The theory of motives was concieved by A. Grothendieck in the 60's with the purpose to study (i.e. prove) the Weil conjectures in the zeta function of algebraic varieties. I plan to start by explaining some background material such as the notion of a Weil cohomology theory and the definition of correspondences. Weil conjectures would be a consequence of 2 stronger statements of topological nature: the so-called Standard Concejectures. I plan to explain them and then head to the defintion of a pure Chow motive.

The definition of the Grothendieck groups attached to an algebraic scheme was motivated by the expectation to generalize the classical Riemann-Roch theorem in the "relative case." I plan to cover the definition and main functorial properties of these groups in the first part of the course, and then explain the statement of the Grothendieck-Riemann-Roch theorem. If time permits, I will mention further generalizations of these ideas developed by Baum, Fulton, and MacPherson.

This course will begin by reviewing the geometry of curves over finite fields (especially linear systems, Jacobians and zeta functions). We will then discuss the application of curves to the construction of codes and cryptosystems. Though some familiarity with number theory and algebraic geometry will be helpful, this course should be accessible to beginning graduate students.


MAT 1196HF (MAT 445H1F)
REPRESENTATION THEORY
F. Murnaghan

A selection of topics from: Representation theory of finite groups, topological groups and compact groups. Group algebras. Character theory and orthogonality relations. Weyl's character formula for compact semisimple Lie groups. Induced representations. Structure theory and representations of semisimple Lie algebras. Determination of the complex Lie algebras.


MAT 1302HS (APM 461H1S/CSC 2413HS)
COMBINATORIAL METHODS
S. Zoble
A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.


MAT 1303HF (CSC 2406HF)
TRIPLE SYSTEMS
E. Mendelsohn

Triple systems have their origins in algebraic geometry, statistics, and recreational mathematics. Although they are the simplest of combinatorial designs, the study of them encompasses most of the combinatorial, algebraic, and algorithmic techniques of combinatorial design theory.

Topics to be covered: Fundamentals of design; existence of triple systems; enumeration; computational methods; isomorphism and invariants; configuration theory; chromatic invariants; resolvability; directed, Mendelsohn, and mixed triples systems.


MAT 1340HF (MAT 425H1F)
DIFFERENTIAL TOPOLOGY
A. Khovanskii

Smooth manifolds, smooth mappings and Sard theorem, differential forms and deRham cohomology, mapping degree and Pontryagin's construction, Euler characteristic, Hopf theorem and Lefschetz fix points theorem. Applications of smooth topology to algebra and calculus.

Reference::
J. Milnor, Topology from the differential viewpoint
Additional Reference::
R. Boot, Tu: Differential forms in algebraic topology


MAT 1344HF
INTRODUCTION TO SYMPLECTIC GEOMETRY
Y. Karshon

We will discuss a variety of concepts, examples, and theorems of symplectic geometry and topology. These may include, but are not restricted to, these topics:

• Review of differential forms and cohomology;
• Symplectomorphisms; Local normal forms;
• Hamiltonian mechanics; Group actions and moment maps;
• Geometric quantization; A glimpse of holomorphic techniques.

This is a course on the main notions and basic facts of symplectic topology. The topics to be covered include: symplectic and contact spaces, Morse theory, generating functions for symplectomorphisms, symplectic fixed point theorems (Arnold's conjectures), and invariants of legendrian knots. We also plan to touch on almost complex structures, groups of symplectomorphisms, definitions of symplectic capacities and Floer homology. In the second part of the course we are going to cover various constructions of completely integrable systems in finite and infinite dimensions. An acquaintance with basic symplectic geometry (e.g. covered by MAT 1344HF, Introduction to Symplectic Geometry) is advisable.

References:
S.Tabachnikov: Introduction to symplectic topology (Lecture notes, PennState U.)
D.McDuff and D.Salamon: Introduction to symplectic topology (Oxford Math. Monographs, 1998)
A.Perelomov: Integrable systems of classical mechanics and Lie algebras (Birkhauser, 1990)


MAT 1404HF (MAT 409H1F)
SET THEORY
W. Weiss

We will introduce the basic principles of axiomatic set theory, leading to the undecidability of the continuum hypothesis. We will emphasize those aspects which both provide a basis for further study and are most useful in applications to other branches of mathematics.

Notes for the course will be available on a website.

Optional Reference Textbooks:
W. Just and M. Weese: Discovering Modern Set Theory, I and II, AMS.
K. Kunen: Set Theory, Elsevier. T. Jech: Set Theory, Academic Press


MAT 1448HF
TOPICS IN GENREAL TOPOLOGY
F.D. Tall

The topics will depend to some extent on the background of the students and how many undergraduates enroll. Topics will include compactness, convergence, paracompactness, and metrization, but at a deeper and more modern level than in typical analysis courses, with attention to counterexamples. There will also be some attention to set-theoretic aspects, but no significant set theory background will be assumed. A number of prospective 4th year students have expressed interest in a Moore-method (otherwise known as "discovery method" or "research at your own level") topology course; if a sufficient number of such students actually enroll, I will run the course that way, but will also give a broad range of survey lectures so that students can be exposed to more material than they would get in a pure Moore-method course.

I am flexible re prerequisites. A standard undergraduate introduction to point-set topology such as MAT 327 will do, as will the graduate Topology Problems course.


MAT 1450HS
SET THEORY: FORCING AXIOMS
S. Todorcevic

This will be an introduction into the study of strong Baire category theorems, starting from the weakest one MA and ending with the maximal one MM. Some bounded forms of these axioms will also be considered as well as their connection with principles of generic absoluteness and &Omega -logic.


MAT 1501HS
GEOMETRIC MEASURE THEORY AND CALCULUS OF VARIATIONS
R. Jerrard

• overview of geometric measure theory: motivations and basic ideas
• review of general measure theory
• area and coarea formulas
• rectifiability: definition and equivalent characterizations
• currents and rectifiability theorems
• applications to Ginzburg-Landau equations, time permitting

Asymptotic series. Asymptotic methods for integrals: stationary phase and steepest descent. Regular perturbations for algebraic and differential equations. Singular perturbation methods for ordinary differential equations: W.K.B., strained coordinates, matched asymptotics, multiple scales. (Emphasizes techniques; problems drawn from physics and engineering.)

Reference:
Carl M. Bender and Steven A. Orszag: Advanced Mathematical Methods for Scientists and Engineers.


MAT 1508HS
NONLINEAR SCHOROEDINGER EQUATIONS
J. Colliander

1. course overview: issues in study of nonlinear Schroedinger (NLS)
2. review of harmonic analysis, function spaces, embeddings
3. review of finite-d Hamiltonian dynamics
4. initial value problem
   1. structural properties
   2. linear estimates
   3. local well-posedness theory
   4. global well-posedness theory
5. long-time behaviour of solutions
   1. scattering
   2. solitons
   3. blow-up
   4. turbulence?

The geometry of Lorentz manifolds. Gravity as a manifestation of spacetime curvature. Einstein's equation. Exact solutions: Schwarzschild metrics and black holes, Applications: bending of light, perihelion precession of Mercury. The Cauchy problem and initial value formulation. The constraint equations for space-like hypersurfaces. The structure of asympototically flat 3-manifolds: ADM mass, positive mass theorem, penrose inequality, center of mass, topology of asymptotically flat three manifolds.

Textbooks:
R. Wald: General Relativity.
O'Neill: Semi-Riemannian Geometry.


MAT 1723HF (APM 421H1F)
FOUNDATIONS OF QUANTUM MECHANICS
R. Jerrard

• physical background
• structure of wuantum mechanics: states and observables
• the uncertainty principle
• dynamics: existence of dynamics, conservation laws
• exact solution of the Schroedinger equation in some special cases
• spectral theory
• introduction to scattering
• introduction to many particle systes and to density matrices.

Mirror symmetry plays a central role in the study of geometry of string theory. In mathematics, it reveals a surprising connection between symplectic geometry and algebraic geometry. In physics, it provides a conceptual guide as well as powerful computational tools, especially in compactifications to four-dimensions.

Outline:

1. Background: Supersymmetry and homological algebra
               Non-linear sigma models (NLSM)
               Landau-Ginzburg models
               topological field theory and topological strings
2. Linear sigma models, moduli space of theories
3. Mirror Symmetry
4. Mirror Symmetry involving D-branes
    * B-branes in NLSM - holomorphic bundles, coherent sheaves
    * B-branes in LG models - level sets, matrix factorizations
    * A-branes in NLSM - Lagrangian submanifolds, Floer homology
    * A-branes in LG models - vanishing cycles and Picard-Lefschetz monodromy


References: The course does not follow a textbook but the following may
            be useful.
1. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa,
   R. Vakil and E. Zaslow, ``Mirror Symmetry'' Clay Mathematics Monographs
   Vol 1 (AMS, 2003).
2. P. Deligne, P. Etingof, D. Freed, L. Jeffrey, D. Kazhdan, J. Morgan,
   D. Morrison, E. Witten, ``Quentum Fields and Strings: A Course for
   Mathematicians'' (AMS 1999).

We will study notions of complexity, which were pioneered in theoretical computer science, as applied to traditional topics in theoretical and applied mathematics, such as Hilbert's Nullstellensatz, Newton's method, linear programming, Julia sets and Mandelbrot sets.

Textbook:
Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale: Complexity and Real Computation, Springer-Verlag, 1998, ISBN 9 780387 982816


MAT 1844HF
UNIFORMIZATION THEOREM AND ITS APPLICATIONS IN HOLOMORPHIC DYNAMICS
M. Lyubich

The classical Uniformization Theorem is one of the most profound results in mathematics. It asserts that any simply connected Riemann surface is conformally equivalent to either the Riemann sphere, or to the complex plane, or to the hyperbolic plane. We will first discuss different proofs of this theorem and then will give some of its numerous applications to holomorphic dynamics (an active field of research that studies iterates of rational endomorphisms of the Riemann sphere).


MAT 1845HF
TOPICS ON HOMOCLINIC BIFURCATION, DOMINATED SPLITTING, ROBUST TRANSITIVITY AND RELATED RESULTS
E. Pujals

For a long time it has been a goal in the theory of dynamical systems to describe the dynamics of "big sets" (generic or residual, dense, etc.) in the space of all dynamical systems. It was briefly thought in the sixties that this could be realized by the so-called hyperbolic ones: systems with the assumption that the tangent bundle splits into two complementary subbundles that are uniformly forward (respectively backward) contracted by the tangent map. Under this assumption, it is completely described the dynamic of f from a topological and statistical point of view. Nevertheless, uniform hyperbolicity was soon realized to be a property less universal than it was initially thought. Roughly speaking, it was showed that two kind of different phenomena can appear in the complement of the hyperbolic systems:

• nonhyperbolic dynamics with the property that the limit set of any perturbation of the initial system can be decomposed into a finite number of closed transitive sets.
• dynamics that generically exhibit infinitely many disjointed transitive sets obtained as a consequences of homoclinic bifurcation.

These new results naturally pushed some aspects of the theory on dynamical systems in different directions:

1. The study of the dynamical phenomena obtained from homoclinic bifurcations.
2. The characterization of universal mechanisms that could lead to robustly (meaning any perturbation of the initial system) nonhyperbolic behavior.
3. The study and characterization of isolated transitive sets that remain transitive for all nearby system (robust transitivity).
4. The dynamical consequences that follows from some kind of the dynamics over the tangent bundle, weaker than the hyperbolic one.

Along the course we will deal with the previous topics discussing the following subjects:

1. Generic dynamics for surface maps: hyperbolicity versus homoclinic tangencies.
2. Generic dynamics for maps in higher dimension: hyperbolicity, homoclinic tangencies and heteroclinic cycle. The homoclinic bifurcation as the universal mechanisms that could lead to robust nonhyperbolic dynamics.
3. The unfolding of homoclinic tangencies and heteroclynic cycles.
4. Examples of robust transitive systems.
5. Partial hyperbolic systems and heteroclinic bifurcations.
6. Robust transitivity and its relation with the dominated splitting.
7. Dynamical consequences from partial hyperbolic systems and dominated splitting.
8. The case of vector fields.

Introduction to the basic mathematical techniques in pricing theory and risk management: Stochastic calculus, single-period finance, financial derivatives (tree-approximation and Black-Scholes model for equity derivatives, American derivatives, numerical methods, lattice models for interest-rate derivatives), value at risk, credit risk, portfolio theory.


MAT 1900Y/1901H/1902H
READINGS IN PURE MATHEMATICS

Numbers assigned for students wishing individual instruction in an area of pure mathematics.


MAT 1950Y/1951H/1952H
READINGS IN APPLIED MATHEMATICS

Numbers assigned for students wishing individual instruction in an area of applied mathematics.


STA 2111HF
GRADUATE PROBABILITY I
J. Rosenthal

A rigorous introduction to probability theory: Probability spaces, random variables, independence, characteristic functions, Markov chains, limit theorems.


STA 2211HS
GRADUATE PROBABILITY II
J. Rosenthal

Continuation of Graduate Probability I, with emphasis on stochastic processes: Poisson processes and Brownian motion, Markov processes, Martingale techniques, weak convergence, stochastic differential equations.

COURSES FOR GRADUATE STUDENTS FROM OTHER DEPARTMENTS

(Math graduate students cannot take the following courses for graduate credit.)

(These courses are used as reading courses for engineering and science students in need of instruction in special topics in theoretical mathematics. These course numbers can also be used as dual numbers for some third and fourth year undergraduate mathematics courses if the instructor agrees to adapt the courses to the special needs of graduate students. A listing of such courses is available in the 2003-2004 Faculty of Arts and Science Calendar. Students taking these courses should get an enrolment form from the graduate studies office of the Mathematics Department. Permission from the instructor is required.)

COURSE IN TEACHING TECHNIQUES

The following course is offered to help train students to become effective tutorial leaders and eventually lecturers. It is not for degree credit and is not to be offered every year.

The goals of the course include techniques for teaching large classes, sensitivity to possible problems, and developing an ability to criticize one's own teaching and correct problems.

Assignments will include such things as preparing sample classes, tests, assignments, course outlines, designs for new courses, instructions for teaching assistants, identifying and dealing with various types of problems, dealing with administrative requirements, etc.

The course will also include teaching a few classes in a large course under the supervision of the instructor. A video camera will be available to enable students to tape their teaching for later (private) assessment.